Combinatorial Proofs of Some Identities for Nonregular Continued Fractions

<table>There are no fault-free elements of <svg height="15.65" id="M137" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 73.012497 15.65" width="73.012497" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.363)"><path d="M646 468q0 -91 -56.5 -143t-138.5 -62q-23 -3 -45 -1l-74 19v-157q0 -63 15.5 -77t83.5 -19v-28h-393v28q62 5 76 19.5t14 77.5v400q0 63 -14 77.5t-76 19.5v28h375q112 0 169 -41q64 -49 64 -141zM550 464q0 77 -42.5 115.5t-108.5 38.5q-44 0 -56 -12q-11 -8 -11 -47
v-245q31 -12 73 -12q59 0 102 38t43 124zM290 131v390q0 62 -7.5 79.5t-30.5 17.5q-24 0 -31.5 -17.5t-7.5 -79.5v-390q0 -62 7.5 -79.5t31.5 -17.5q23 0 30.5 17.5t7.5 79.5z" id="x2119"></path></g>
<g transform="matrix(.012,0,0,-.012,11.675,15.45)"><path d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2
q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" id="x1D45B"></path></g>
<g transform="matrix(.017,0,0,-.017,22.05,11.363)"><path d="M528 54l-36 -38l-198 201l-198 -201l-36 38l197 200l-197 201l36 38l198 -202l198 202l36 -38l-197 -201z" id="xD7"></path></g><g transform="matrix(.017,0,0,-.017,35.785,11.363)"><path d="M794 -138l-9 -21q-74 -1 -142 15q-79 19 -254 130q-139 7 -231 87q-52 44 -83 107t-31 135q0 145 103 247.5t274 102.5q145 0 249.5 -93.5t104.5 -240.5q0 -126 -73 -212.5t-179 -116.5q137 -85 271 -140zM673 302q0 152 -78 239.5t-201 87.5q-50 0 -84.5 -74t-34.5 -209
q0 -265 118 -317l66 18l31 -19q76 19 129.5 90t53.5 184zM298 54q-66 94 -66 292q0 178 50 256q-62 -32 -99 -98t-37 -162q0 -98 41 -174t111 -114z" id="x211A"></path></g>
<g transform="matrix(.012,0,0,-.012,49.688,15.45)"><path d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2
q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" id="x1D45B"></path></g><g transform="matrix(.012,0,0,-.012,55.661,15.45)"><path d="M535 230h-483v50h483v-50z" id="x2212"></path></g><g transform="matrix(.012,0,0,-.012,62.645,15.45)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg> for the case <svg height="7.9499998" id="M138" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 8.6625004 7.9499998" width="8.6625004" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,7.675)"><path d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2
q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" id="x1D45B"></path></g>
</svg> is odd.</table>

International Journal of Combinatorics

fig5

Figure 5

Figure 5: Combinatorial Proofs of Some Identities for Nonregular Continued Fractions 